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Turbulent Aggregation of Alumina in Water andn-Heptane
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
202, 238–250 (1998)
CS985436
Turbulent Aggregation of Alumina in Water and n-Heptane Hubert Saint-Raymond, Fre´de´ric Gruy, and Michel Cournil 1 Ecole des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint Etienne Cedex 2, France Received April 15, 1997; accepted January 23, 1998
THEORETICAL BACKGROUND Aggregation of alumina powder in water and n-heptane is followed by turbidimetry in situ. The experiments are carried out under turbulent conditions. They are discussed in the framework of a model which takes into account the hydrodynamic and physicochemical aspects of aggregation. A good agreement between measured and predicted turbidity is found provided the aggregates are considered as noncompact and liable to undergo fragmentation. Recent models of aggregation and fragmentation of fractallike clusters are successfully tested. The aggregate morphology seems to depend on the nature of the liquid medium, however, not on the stirring rate. q 1998 Academic Press Key Words: aggregation; fractal; turbulence; breakage; turbidimetry; light scattering.
Turbulence
In this work, we adopt the classical description of turbulence as presented in (1–4). At any instant and in any place of the turbulent flow, eddies are supposed to appear and to take part in the energy dissipation. Energy is supplied to the flow by an external source, such as mechanical stirring. Energy is transferred from the largest eddies to the smallest eddies in which it is dissipated through viscous interactions. The size of these smallest eddies is the Kolmogorov microscale h; from dimensionless considerations h is expressed as a function of the kinematic viscosity n and the energy dissipation rate per unit mass em in the form
INTRODUCTION
hÅ
Aggregation is one of the most complex steps in powder technology; this complexity is due to the variety of phenomena or aspects involved: hydrodynamics of suspensions, interactions between particles, and morphology of the aggregates. In spite of the existence of several, valuable models applicable to the separate aspects of the phenomenon, so far, very few comprehensive theoretical approaches are able to describe the turbulent aggregation of polydisperse particles in physicochemical and hydrodynamic interaction, particularly if the so-formed aggregates are not compact and can break up. This is the task that we propose to undertake in this paper, using results from a turbidimetric study of aggregation of alumina powders in water and in n-heptane. This paper is organized as follows: after a review of the theoretical background connected with aggregation, we present the experimental data available on alumina; then, the different steps of the model, each corresponding to a theoretical aspect, are discussed and adapted to the present case. Last, theoretical predictions are compared to the experimental results, and a global interpretation is proposed.
1
To whom correspondence should be addressed.
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n3 em
1/4
.
[1]
The velocity gradient gg in each eddy is proportional to ( em / n ) 1 / 2 . The resulting shear stress is ts Å mgg .
[2]
In this expression, m denotes the dynamic viscosity. Interaction Between Solid Particles
Aggregation is the consequence of a collision between particles. The mechanism that brings particles into close proximity results from the hydrodynamics of the suspension; however, the trajectories of the particles just before their encounter, the collision efficiency, and the aggregate cohesion depend on the different interactions between particles. London–Van der Waals Attractive Interactions These forces have their origin in interactions between instantaneous induced dipoles. For two spheres of radii ai and aj separated by a distance H, an attractive interaction potential VA can be calculated (5). In fact, a more correct expression is obtained by taking into account a ‘‘retardation’’ effect (6) due to propagation of electromagnetic waves in a dielec-
238
0021-9797/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.
S D
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TURBULENT AGGREGATION OF ALUMINA
tric medium. Schenkel and Kitchener (7) have proposed an empirical correlation to calculate the attractive potential as a function of the geometrical parameters (ai , aj , H) and of A, the Hamaker constant of the particle in its medium. Double-Layer Repulsive Interactions A solid particle located in an electrolyte generally takes a surface charge and is surrounded by a layer of dissolved ions. Several models exist to describe this double layer (5, 8, 9). The zeta potential is the only characteristic parameter of this double layer which is experimentally available; it is currently assimilated to the surface potential c0 in the case of dilute liquid media. When two particles get into contact, the interactions of their respective double layers result in a repulsive potential VR . In this paper, only experiments for which the double-layer interactions were zero or negligible are presented. For this reason, we do not give further details on this subject. Hydrodynamic Interactions The problem is to describe the conditions of collision of two interacting particles brought into close proximity by a viscous fluid. Following the pioneering works of Smoluchowski ( 10 ), several authors ( 11, 12) expressed the Brownian aggregation flow Jij ( R) of particles of radius aj which collide with the sphere of radius R surrounding a reference particle of radius ai : Jij Å 04pR 2G(R) 1
F
(Di / Dj )
ÌNj Nj ai / aj ÌVT / ÌR 6pm ai aj ÌR
G
.
[3]
G(R), the corrected mobility coefficient, is a function of R, ai , and aj ; Di is the Brownian diffusion coefficient of a sphere of radius ai Di Å
k BT , 6pmai
[4]
where kB is the Boltzmann constant, T is the temperature, Nj is the number of particles of radius aj per unit volume, and VT is the total interaction potential. The effect of these interactions is characterized by the stability factor Wij , defined as Wij Å
J 0ij (R) , Jij ( R)
Wij which takes into account the attractive and repulsive interactions and also the effect of the viscosity via G(R). The models derived from Eq. [3] assume that aggregation takes place through Brownian motion. Brownian aggregation is important for submicrometer particles; however, when their size becomes larger, the particles are submitted to the action of the flow and other aggregation mechanisms take place. More recent studies (13–16) take into account the different types of interactions in a shear flow. Their authors introduce the capture efficiency coefficient aij between two particles of radius ai and aj ; aij is the ratio of the calculated particle collision flow to the flow value predicted by Smoluchowski ( aij Å W 01 ij ). The model of Van de Ven and Mason (13) includes attractive, repulsive, and hydrodynamic interactions in the case of equally sized particles, whereas the model of Higashitani et al. (15, 16) can be applied to any pair of spherical particles, in the case of attractive and hydrodynamic interactions only. The case of aggregation in turbulent flows has been little investigated; two tracks have been explored so far. For particles smaller than the Kolmogorov microscale, aggregation is assumed to take place in the smallest eddies under local shear flow conditions. This makes possible the use of the previous models (15–19). Most recently, two of us (20) have proposed the use of the Spielman equation (Eq. [3]) by replacing the Brownian diffusion coefficient Di with the Levich (21) turbulent diffusion coefficient Dturb . We proposed an analytical expression for aij in the most general case of interactions and radii values in a turbulent flow. Collision Efficiency for Noncompact Aggregates
The morphology of the aggregates depends on both the physicochemical and the hydrodynamic conditions of their formation as well as on their intrinsic mechanical properties. However, the aggregation dynamics also depend on the morphology of the colliding particles. Recent experiments have shown that aggregates have a fractal structure (22, 23). An aggregate containing i primary particles of radius a1 is characterized by the fractal dimension Df , the outer radius ai , and the hydrodynamic radius aH . Because the structure of the aggregates is nonuniform, their volume density f(r) depends on the distance r from the center of mass of the aggregate, where the average volume density is denoted fU . These different characteristics are linked by the relations (24, 23)
[5]
ai Å a1
where J 0ij (R) denotes the Jij ( R) value in the absence of interactions. Spielman (12) has proposed a determination of
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f(r) Å
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SD SD i S
1/D f
S r Df 3 a1
[6] D f 03
[7]
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SAINT-RAYMOND, GRUY, AND COURNIL
SD
fU Å S
ai a1
D f 03
,
[8]
where S is a structure factor. The authors differ in the way they express the ratio between the hydrodynamic radius and the outer radius; this ratio is assumed to be constant by (22, 25), whereas, according to (23), it depends on the aggregate volume density. However, for fractal dimensions close to 2, the two models are in agreement. From computer simulations, Gmachowski (24) has found the following relation between S and Df : S É 0.42 Df 0 0.22.
[9]
We present here the procedure used by Kusters (23, 26) (shell-core approach) to calculate the collision efficiency aij between two porous aggregates (i ¢ j ú 1). (i) Determination of the permeability of the bigger particle (26): 30 ki Å
9 fU 2
1/3
/
S
9 fU 2
9fU 3 / 2fU
05 / 3
D
5/3
0 3fU
2
2a 21 ,
[10]
CU
where CV is an average shielding coefficient. (ii) Determination of the Debye shielding ratio of this particle: jÅ
ai . k 1/2
[11]
(iii) Determination of the collision efficiency (Kusters model): a iK, j as a function of j; from the results given in (26), the following relations can be derived after curvefitting for aj ú 0.1: ai
a iK, j ( j ) Å 1 0 tanh 0.1 j 1.35 ,
[12a]
aj õ 0.1: ai
a iK, j ( j ) Å 1 0 tanh 0.18 j 0.4 .
[12b]
(iv) Determination of the collision efficiency a iH, j (Higashitani model) for compact aggregates (15, 16). (v) Comparison between a iK, j and a iH, j and choice of the ai , j value. The shell-core model takes into account the porous fractal structure of the aggregates but ignores the interactions other
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than hydrodynamic, whereas, within the Higashitani approach, hydrodynamic and attractive interactions are considered between compact aggregates. Somewhat to take advantage of the most interesting aspects of each model, it is proposed in (23) that ai , j be chosen in the following way: if a iK, j õ a iH, j , then ai , j Å a iH, j , else ai , j Å a iK, j . Break-Up of Aggregates
In the aggregation processes, a maximum aggregate size is almost always observed (19, 26, 28). This can be due to several reasons: breakage, collision efficiency becoming zero beyond a critical size, or settling which could remove the largest aggregates from the system. The second cause is unlikely in the case of noncompact aggregates, and the third one requires particle sizes much larger than the maximum size reached experimentally. Breakage is certainly the main cause of this limit in size. The occurrence of breakage depends on the balance between the desaggregation effects due to the action of the fluid and the overall cohesion of the aggregate due to the interactions between primary particles. The hydrodynamic effects are of different nature depending on whether the aggregate is larger or smaller than the Kolmogorov microscale. Only the latter case is compatible with the experimental conditions of this study. It corresponds to a shear stress originating from the local velocity gradient and acting on the aggregate. The authors do not agree on the way to express the competition between the desaggregation and the cohesion effects. According to the models, the relevant parameter is either the ratio Ec /Et or the ratio s / ts ; Ec and Et are, respectively, the aggregate cohesion energy and the turbulent energy acting on this aggregate; s is the mean mechanical strength of the aggregate; and ts is the mean shear stress (22, 26, 27, 29). The breakage rate is proportional respectively to gg e 0E c /Et and to gg e 0s/ts . Another discussion is relative to the size of the fragments produced by breakage. Two cases are currently envisaged: • Erosion of single or small groups of particles from the aggregate surface (22). • Production of fragments of comparable sizes (26, 27).
In all cases, the breakage rate depends on the hydrodynamic conditions of the flow, via em and m, and on the characteristics of the aggregates—outer radius, fractal dimension, primary particle radius, and cohesion force between two primary particles. This force F is expressed in the form FÅ
Aa1 . 12H 20
[13]
H0 is the separation between two primary particles in the aggregates.
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Aggregation Kernels
The variation of the volume density Ni of aggregates Ai versus time t is given by the population balance equation in its discrete form (30) for i ú 1:
dNi 1 Å dt 2
∑ Kj,i0j Nj Ni0 j 0 ∑ Ki ,k Ni Nk j Å1,i01
0
i Å 1:
1 2
kÅ1,`
∑ K if, j Ni / ∑ K j,if Nj j Å1,i01
[14]
j úi
dN1 f Å 0 ∑ K1,k N1 Nk / ∑ K j,1 Nj , dt kÅ1,` j ú1
[15]
where Kij is the aggregation kernel [i-mer / j-mer r (i / j)-mer] and K if, j the breakage kernel [i-mer r j-mer / (i 0 j)-mer]. According to the type of flow and the particle size, Brownian kernels or turbulent kernels can be considered. The relevant parameters are b1 , the ratio of the particle size to the Kolmogorov microscale, and b2 , the ratio of the particle size to the Levich critical diameter dc (21). For b1 ú 1, models of highly turbulent suspensions should be applied (31). For b1 õ 1 and b2 ú 1, the turbulent models (which in fact assume a local shear flow in the Kolmogorov’s eddies) can be used. For b1 õ 1 and b2 õ 1, the Brownian contribution is no more negligible. In this case, Adachi et al. (32) propose that Kij be expressed as the sum of two contributions Kij Å (Kij )Br / (Kij )turb
[16]
in which (Kij )Br is the Brownian kernel given by Smoluchowski and possibly corrected by the stability factor Wij (equal to 2 in the experimental work of Adachi) and the turbulent kernel (Kij )turb is derived from one of the turbulent models presented earlier. For instance, in the case of particles smaller than the Kolmogorov microscale, the following expression is currently used: (Kij )turb Å
4 3
S D SD 3p 10
1/2
em n
1/2
d (ai / aj ) 3 aij . [17]
This relation comes from the Saffmann-Turner approach (17); em is the mean energy dissipation rate; d is a correction coefficient which is introduced to take into account different deviations from this ideal model. Many expressions are found in the literature for the mean value of em . For instance, Baldi et al. (33) give Np v 3 D 5s , V
[18]
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eU m Å
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where Np is the power number, Ds is the stirrer diameter, v is the rotation rate of the stirrer, and V is the volume of the suspension. This type of expression should be used with precaution because, except for highly turbulent media, em is not uniform in a stirred vessel. In particular, Wu and Patterson (34) show that, in a baffled stirred tank with a sixblade turbine, the dissipation in the impeller region is about 15 times as much as that in other parts of the tank; the fluid in this region represents only 9% of the total. The consequence of the turbulence heterogeneity can be taken into account via d [ for instance, d Å 0.7 in (26)]. Particles in an unstirred medium can form aggregates while settling (35, 36). Different aggregation kernels have been proposed according to the particle size concerned. For small sizes (ai õ 50 mm), the terminal velocity of a particle is given by the Stokes law and is proportional to a 2i ; hence, the expression of the aggregation kernel for gravitational settling is Kij Å
2p ( rs 0 rL )g (ai / aj ) 2Éa 2i 0 a 2j É . m 9
[19]
In this relation, rs and rL are the respective densities of the solid and liquid phases; g is the gravity acceleration. Turbidimetry
General Scope In this work, aggregation of alumina powders has been studied using an in situ turbidimetric sensor. This device allowed us to record continuously a signal directly dependent on the particle size distribution (PSD) in the reactor and so we avoid two drawbacks: • Sampling from the suspension, which could damage the aggregates and definitely introduces systematic counting errors, in particular concerning the finest particles which are difficult to collect. • A bypass through the measurement cell of a commercial particle sizer, with again the risk of damaging the aggregates by pumping and definitely by uncontrolled hydrodynamical disturbances.
The turbidity of a polydisperse suspension of spheres of diameter d is linked to the PSD by the Mie theory (37, 38), t( l ) Å
*
`
Csca f ( d)dd,
[20]
0
where t( l ) denotes the turbidity at wavelength l; f ( d), the population density; and Csca , the Mie scattering cross-section. The inverse problem (i.e., the determination of the PSD from the measured turbidity spectrum) is acknowledged to
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FIG. 1. Schematic diagram of the experimental setup.
be an ill-conditioned mathematical problem which can be solved by the use of constrained least-squares methods (39). In recent works, we defined the interests and limits of such a method in particle sizing (40, 41). For instance, a low volume fraction f of solid is required, generally f õ 10 04 for micrometer particles. Due to its ill conditioning the inverse problem can be solved in good conditions only if the main part of the sample belongs to the diameter range 0.1– 10 mm. Nevertheless, the direct calculation of turbidity spectra, as expressed by relation [20], is valid for all particle diameters.
Most recently, a procedure of determination of light scattering and extinction by an ensemble of fractal clusters was proposed by Khlebtsov (42) and validated for a system consisting of aggregating polystyrene latex particles. However, due to the size of the primary particles (90 nm) , the approximation of Rayleigh-Gans —here approximately equivalent to a1 ! l — was used by Khlebtsov in its work and the aggregate scattering cross-section Csca,R0G was calculated in this theoretical framework. For larger primary particles, not too far from the Rayleigh-Gans domain, it has been suggested (43) to calculate Csca , the actual value of the scattering cross-sections by correcting Csca,R0G in the following way: C 1sca C 1sca,R0G
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Experimental Setup, Experimental Procedure and Products
(i) Measurement set of turbidity. The light source is a 75-W high-pressure Xe–Hg arc lamp which emits a polychromatic light in the UV–visible range (190–800 nm). Light is passed to the sensor window through optical fibers
[21]
In this relation, C 1sca and C 1sca,R0G are both the scattering cross-sections of a primary particle; however, the former has
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The experimental set-up used in this study can be divided into three parts (Fig. 1).
Turbidity of Aggregates
Csca Å Csca,R0G
been calculated using the rigorous Mie theory, whereas the latter is obtained from the Rayleigh-Gans approximation. The Khlebtsov procedure has been conceived for aggregates of many particles. For doublets, for instance, the notion of fractal dimension has no meaning, of course; however, the optical basis of the Csca calculation can be kept (44).
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FIG. 2. Top view of the stirred vessel (fitted with a turbidimetric sensor ) .
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FIG. 3. Turbidity variation throughout a typical experiment.
600 mm in diameter. The window length is the optical path which ranges from 0.1 to 17 mm. Each of the sensor window ends is fitted with a convergent lens which ensures the beam collimation in the measurement zone. After the light beam has crossed the sensor window, scattering by fine particles results in the attenuation of its intensity. The transmitted light is again passed via optical fibers to a spectrophotometer which is equipped with a monochromator (holographic diffraction grating) and a light detector consisting of a 35 photodiode array. This setup can deliver continuously the turbidity spectrum of the suspension. (ii) Reactor. The reactor (Fig. 2) is a glass cylindrical vessel with diameter and height both equal to 10 cm. It is double-jacketed to ensure the temperature control. The turbidity sensor is located at 3 cm above the bottom. Stirring is operated by a four-blade 457 impeller that is 4.6 cm in diameter and located at the same height as the sensor. To ensure a good dispersion of the solid phase at the beginning of each experiment, a ultrasonic probe can be used. (iii) Data acquisition and processing set. Each available turbidity spectrum is in fact the average of 20 spectra recorded in the total time of 3 s. This allowed us to follow the evolution kinetics of relatively rapid processes over a period of several hours.
FIG. 4. Initial particle size distribution (PSD) of alumina in water.
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243
FIG. 5. Zeta potential of alumina in water at different pH values.
Each experiment consists of five different steps which are represented in Fig. 3: ( i ) Filling the reactor with 500 cm3 of the desired liquid medium; starting thermal control and stirring ( zone I in Fig. 3 ) . (ii) Measuring of the blank turbidity signal (i.e., in absence of solid). (iii) Introducting the solid sample (10–100 mg); natural dispersion in the liquid medium during a 5-min period (zone II in Fig. 3). (iv) Starting the ultrasonic action for a 5 min time (zone III in Fig. 3). (v) Ending the ultrasonics action; this instant is taken as zero time (t 0 in the figures) for the free evolution of the system (zone IV in Fig. 3). In this work, aggregation has been studied on 50-mg samples of alumina powders from Baikowski Company. This alumina
FIG. 6. Turbidity variation with time during aggregation for different wavelengths (50 mg alumina, pH Å 9.5, stirring rate Å 350 rpm).
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FIG. 7. Turbidity spectra at different times during aggregation (50 mg alumina, pH Å 9.5, stirring rate Å 350 rpm).
consists at least of 95% a-alumina with a BET specific surface area of 5.7 m2rg 01 . Figure 4 shows the PSD by number obtained from measurements in a Coulter LS 130 laser particle sizer. Immediately before the PSD determination, the sample was dispersed ultrasonically. A large number of particles have diameters close to 0.2 mm, whereas fewer aggregates with diameters ranging from 1 to 6 mm are observed. This is confirmed by microphotographs. Figure 5 shows the zeta potential values of alumina in water measured on a Pen Kenn Laser Zee Meter at different pH values. High absolute values of zeta potential are currently associated with strong repulsive interactions and consequently to weak aggregation, whereas for low values— say smaller than 20 mV, corresponding to the pH zone 8– 9.5—aggregation is assumed to be easier. Most of the experiments reported here have been performed for pH values belonging to this interval. The zero value of zeta potential
FIG. 8. Reversibility of aggregation (US Å ultrasonics action).
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FIG. 9. Influence of pH on turbidity variation during alumina aggregation in water (350 rpm; l Å 501 nm).
is reached for pH 9. This result is in agreement with the available experimental data (45). Results
The available results from each experiment can be put in two forms: t(t) for different wavelengths (Fig. 6) or t( l ) at different times (Fig. 7). Because little will be drawn from the turbidity spectra, we will comment mainly on t (t) plots obtained at a given wavelength value. Experiments in Water Reversibility of the aggregation process. Figure 8 shows the effect of ultrasonics on aggregation of alumina at pH
FIG. 10. Influence of stirring rate on turbidity variation during alumina aggregation in water (pH Å 9.5; l Å 501 nm).
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sufficient to breakup the aggregates and to recover the initial state. DISCUSSION
Qualitative Discussion
FIG. 11. Influence of stirring rate on turbidity variation during alumina aggregation in n-heptane ( l Å 501 nm).
9.5 and a stirring rate of 350 rpm. After the final t value has been reached, ultrasonics are applied once again. After their action period, the same values of initial turbidity and the same time evolution as previously are observed. These experiments tend to prove that the aggregation of alumina in water is widely reversible: • The aggregates can be destroyed in their elementary components under the action of a sufficient mechanical stress. • The identity of the primary particles is not affected by aggregation, since they can apparently be recovered with the same properties.
The continuous decrease in turbidity as aggregation proceeds can be essentially explained by the decrease in particle number (Eqs. [14], [15], and [20]). The effect of the stirring rate is certainly to increase the frequency of collisions between particles and/or aggregates. The fact that t(t) does not tend asymptotically to zero could be explained by the existence of a maximum aggregate size. This important point will be examined further. In acid (pH õ 6.7) or basic media (pH ú 10), aggregation is very weak. In these two pH zones, the absolute value of zeta potential (Fig. 5) is greater than 30 mV; this corresponds to a strong repulsive interaction. In contrast, in the pH zone, 8.5–9.5, the very low values of zeta potential allow aggregation to occur. The maximum aggregation rate is observed for pH 9.5, that is, for a nonzero zeta potential value, and namely equal to 020 mV. This is hardly understandable in the framework of the classical theories which consider that a nonzero potential results in repulsive forces between particles and reduces aggregation. Because the measurements of zeta potential have performed under experimental conditions very similar to the present ones, the only explanation for our observation is that, in our case, the zeta potential cannot be assimilated to the surface potential, contrary to
Influence of pH. The pH of the aqueous solution has been varied by addition of hydrogen chloride or sodium hydroxide solutions. Figure 9 shows the different t(t) plots obtained at 350 rpm and at 501 nm at different pH values. In acid media, turbidity varies little. In the pH domain 8.5– 9.5, a sharp turbidity drop occurs at early times. For pH values larger than 10, the turbidity varies very slowly. In what follows, we will focus our interest on the experiments performed at the pH value of 9.5, because it corresponds to the most rapid variation in the turbidity. Influence of the stirring rate. The influence of the stirring rate is presented for a pH value of 9.5, a mass of 50 mg, and a wavelength of 501 nm (Fig. 10). Results for other pH values are presented in (46). Experiments in n-Heptane A turbidity decrease with time is observed too for alumina suspensions in n-heptane. The effect of the stirring rate is shown in Fig. 11. In contrast to the behavior observed in water, aggregation in n -heptane is not an reversible phenomenon; in particular, the action of the ultrasonics is not
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FIG. 12. Normalized scattering cross-section of fractal aggregates (fractal dimension Df ; number of primary particles in an aggregate n).
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the usual assumptions. Thus we will consider that the surface potential is zero in n-heptane as well as in water at pH 9.5. Because of this unexpected difficulty and also because no model is available to represent aggregation between porous grains in the presence of repulsive electrostatic interactions, we do not deal in this paper with the interpretation of the influence of pH on aggregation.
Models
• The advantage of a not too small wavelength value is to keep the system not too far off the validity range of the models of light scattering by fractal objects (Rayleigh-Gans domain for the primary particles).
In both respects the choice of the value of 501 nm appears a good compromise. Turbidity spectra at zero time. The PSD of alumina in water as obtained from the Coulter LS130 sizer measurements is bimodal and characterized by two peaks of respective mean diameter and geometrical standard deviation:
Turbidity Turbidity calculations. The solid samples studied here contain individual particles of diameter d1 Å 0.2 mm and larger aggregates, which probably consist of these primary particles. The wavelength range investigated is 300–800 nm. Thus, all the particles and aggregates are outside the Rayleigh-Gans domain (d ! l ); however, they are not very far from its upper validity limit. Consequently, the different scattering cross sections have been calculated according to the models presented in the introduction of this paper: • Mie rigorous theory for the primary particles; • approach of Lips and Levine (44) for the doublets; • corrected Khlebtsov procedure for larger aggregates.
Figure 12 shows the results of the calculations for different values of the fractal dimension and the number of particles in the aggregates. To emphasize the effect of the aggregate fractal structure, the scattering cross section (Csca, f ) has been made dimensionless through normalization by the corresponding scattering cross section of a compact object (Csca,c ) with the same number of primary particles. Using this representation, it clearly appears that a ramified object can scatter much more light than a compact object, and this occurs the more ramified and large it is. Once calculations for these scattering cross-sections have been done, the turbidity of a given population of particles and aggregates can be determined from Eq. [20]. Exploitation of the turbidity data. In the present work, the inversion procedure did not successfully calculate the PSD from the measured turbidity spectrum probably because of the broad PSD observed as early as the beginning of aggregation. However, relation [20] has been widely exploited to calculate the turbidity spectra corresponding to PSD derived from the different aggregation models. Thus validity of the models has been judged from the comparison between predicted and measured turbidity values. These comparisons have been often performed at the value of 501 nm in wavelength for two reasons: • It is one of the wavelength values for which the turbidity drop was always significant (in contrast to other values such as 800 nm).
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dU 1 Å 0.19 mm; s1 Å 0.42 (primary particles) dU 2 Å 1.3 mm; s2 Å 0.25 (aggregates). The ratio of the number of aggregates to the number of primary particles is: N2 /N1 Å 5.6 1 10 04 . Two problems arise about these data; the problems concern the representativeness of the data to describe the initial state of the aggregating powder: j
The PSD characteristics are obtained from a commercial software which probably ignores the refinements of fractal structures. j The measurements have not been performed in line, although the solid samples have been dispersed ultrasonically. The former reservation can be easily objected to because, due to the low number of aggregates in the sample, the calculated turbidity spectrum depends only slightly on the fractal dimension. Concerning the second point, we have found that the calculated turbidity spectrum is in agreement with the experimental data within 10% accuracy. Thus we can conclude in favor of the representativeness of the sizer measurements. Because no experimental PSD of alumina in n-heptane was available, we estimated it through curve-fitting of the experimental turbidity spectrum to obtain the bimodal distribution: dU 1 Å 0.19 mm; s1 Å 0.32 (primary particles) dU 2 Å 1.3 mm; s2 Å 0.25 (aggregates) and
N2 Å 1.13 1 10 03 . N1
The corresponding calculated turbidity spectrum is in agreement with the measured spectrum within 5% accuracy. The larger proportion of aggregates is likely due to stronger cohesion forces and consistent with the not completely reversible aggregation observed in n-heptane. Aggregation Model: Main Assumptions Models of aggregates. The aggregate fractallike structure is described using the relations [6] – [9]; it is character-
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ized by the parameters d1 (d1 Å 0.2 mm) and Df . The fractal dimension has been assumed constant throughout an aggregate, whatever its size, and over a whole experiment. However, it has been considered as a priori depending on the experimental conditions, in particular, stirring rate and nature of the dispersing medium. The problem of the determination of H0 will be discussed later. Aggregation kernels. The turbulent kernels have been calculated from relation [17 ] using the method proposed by Kusters and recalled earlier. The nonuniformity of the energy dissipation rate in the reactor has been taken into account using the same value of the correction coefficient as we did. This appeared to us to be a reasonable assumption, considering the analogies between the two reactors. In these calculations, the power number Np has been taken equal to 1.2, a value commonly assumed for a four-blade 457 impeller ( 33 ) . The global aggregation kernel (i.e., including the turbulent and Brownian contributions) has been determined following the Adachi relation (Eq. [16]). As we said before, no repulsive interaction has been considered. In our experimental system, the Hamaker constant is equal to 2.67 1 10 020 J in water and 2.38 1 10 020 J in n-heptane; from this value, the Brownian stability factor Wij has been found to be close to 2 in most cases. The kernel of aggregation in a still medium has been calculated using relation [19]. Fragmentation kernel. In this paper, we have successively tested the assumptions of fragmentation of Ayazi Shamlou et al. ( 22 ) ( surface erosion ) and of Kusters ( 26 ) ( breakage in fragments of comparable sizes ) . In each case the corresponding fragmentation kernel K if, j has been calculated. The erosion model has been used without modification; its parameters are H0 and Df . Concerning the second model, we have retained its main assumption, that is, the possibility of breakage in fragments. However, we have envisaged several fracture ratios: i r 2 ( i / 2 ) , i r ( 3 i / 4 / i / 4 ) , . . . and adopted the way of calculation of Ayazi Shamlou et al. ( 22 ) which seemed us to be more satisfactory from a physical point of view. In the same way as in the calculations of the aggregation kernels, we took into account the nonuniformity of the turbulence; consequently, the fragmentation kernel K if, j has been obtained by averaging : gg e 0 s / ts . In the case of fragmentation too, H0 and Df , as well as the fracture ratio, must be estimated.
gated, that is, 0.2–20 mm, may require excessive computation time. To avoid this difficulty, we have adapted a discretization method proposed by Spicer and Pratsinis (47) and known as particularly efficient and rapid in the coagulationfragmentation problems. The size domain is divided in adjacent sections numbered by index i. To the ith section is associated a volume Vi that is the average volume of the aggregates contained in this section, Vi Å
bi / bi01 , 2
where bi is the upper boundary volume of section i; the volume of section i is obtained from the volume of the previous section i 0 1 by the relation Vi Å 2Vi01 (V1 corresponding to the volume of a primary particle). In this study, the discretization in diameter has been achieved using 20 sections. The so-obtained ordinary differential equations system has been solved using the explicit Euler method (which revealed as efficient as the RungeKutta algorithm, currently used in this sort of problems). For each calculated PSD, the corresponding turbidity can be determined using Eq. [20]. The validation of the different models and the estimation of the unknown parameters were carried out from the comparison between experimental and simulated t(t) at a given wavelength, here 501 nm. More
Numerical Methods The evolution of the particle size distribution is simulated by solving the population balance equations [14, 15]. The necessary discretization over the wide size range investi-
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FIG. 13. Calculated turbidity variation with time as a function of the fractal dimension Df of the aggregates (exp.: experimental plot at 300 rpm and l Å 501 nm; aggregation in water).
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precisely, each of these curves has been first made dimensionless by division through the initial turbidity value t(0). Simulation Results for Aggregation in Water
Determination and Discussion of the Relevant Aspects of the Aggregation Process Interest of the fractallike aggregate models. In order to judge the opportuneness of using fractal aggregation models, we present, in Fig. 13, t(t) plots calculated for different values of the fractal dimension Df . No fragmentation has been considered in these simulations. Only the results obtained for the alumina-water system at the stirring rate 300 rpm (Fig. 13) are discussed here. In the same figure, the experimental plot has been reported too. It is quite clear that aggregation proceeds faster when the fractal dimension decreases and that the process experimentally observed is much faster than the prediction derived from a model of nearly compact aggregates (Df Å 2.95). From comparison between the initial parts of the predicted and observed turbidity curves, the fractal dimension Df has been estimated to the value of 2.30. A most interesting feature in these turbidity plots is their nonzero asymptotic value. This apparent stop in the system evolution is currently attributed to the effect of breakage; however, in these simulations, breakage has still not been considered. Although aggregation proceeds, the turbidity signal no longer varies. The origin of this behavior is in the properties of light scattering of the large fractal aggregates. Nevertheless, this effect is not sufficient to explain the height of the final turbidity level; thus, breakage must be considered too. In the absence of stirring, calculations show a very slow change in turbidity, in agreement with the experimental observations. Influence of the break-up phenomenon. We have first examined the influence of break-up via erosion for a stirring rate of 300 rpm, assuming a fractal dimension of 2.30. The best agreement between calculated and experimental curves was found for H0 Å 8 nm. Then, the breakage of the aggregates in fragments of several size ratios has been envisaged. In all cases, the influence of the fracture ratio reveals as weak. In contrast, the fragmentation phenomenon is strongly dependent on H0 , at least, beyond a minimum threshold (to H0 Å 0 nm and H0 Å 0.5 nm, nearly the same plots). This effect is shown in Fig. 14, which is relative to Df Å 2.30. The best agreement with the experimental curve is found for H0 Å 1.5 nm. In several studies of fragmentation, H0 is often considered as close to 1 or 2 nm (22). Consequently, from the comparison between the two fragmentation models, it appears that the value of H0 predicted by the latter (breakage in two fragments of comparable sizes) is more realistic.
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FIG. 14. Calculated turbidity variation with time as a function of the separation H0 between primary particles in the aggregates; fractal dimension Df Å 2.30 (exp.: experimental plot at 300 rpm and l Å 501 nm; aggregation in water).
Identification of the Model Parameters Starting from the previous considerations and estimations, we have determined the values of Df and H0 for which the best agreement is found between the calculated and experimental t(t ) plots. It appears clearly that both the fractal dimension Df of the aggregates and the separation H0 are nearly constant whatever the stirring rate. The best agreement is found for the already determined values H0 Å 1.5 nm and Df Å 2.30. The most recent turbulent aggregation models (15, 16, 32) are globally in agreement with the experimental results provided they take into account two modifications: ( i ) Assumption of fractallike aggregates ( Df É 2.3 ) . This confirms the results of Kusters et al. ( 23 ) . The fractal dimension in a stirred vessel is equal to 2.30 { 0.05, independent of the stirring rate. According to ( 28, 23 ) , this relatively high value of the fractal dimension is indicative of a restructuring of the aggregates due to low interaction forces between primary particles. This restructuring has been also observed by Allain et al. ( 48 ) for aggregation during settling. ( ii ) Aggregates breakage in two fragments of comparable sizes. Simulation Results for Aggregation in n-Heptane
Alumina aggregation in n-heptane differs from aggregation in water on several points:
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• The process seems to be more rapid in n-heptane and particularly at low stirring rate. • Aggregation occurs at a significant rate even in still medium. This points to a mechanism of aggregation through sedimentation. Compared to water, n-heptane is both less dense and viscous, making sedimentation easier. • At low or moderate stirring rate, the turbidity plots present some similarities of shape with the plot obtained for still medium. This could indicate that sedimentation plays an important part in agitated systems too. • For higher stirring rates, this similarity is less visible. This means that the turbulent aggregation mechanisms have now a major influence on the process. • The final plateau of the turbidity plots is lower than previously found. This could indicate a lesser importance of fragmentation, but is quite consistent with the observation of incomplete reversibility of aggregation in n heptane.
As previously stated, simulations have been performed to validate the different models and determine the characteristics of the aggregates. Turbidity plots have been calculated: ( i ) in the case of the still medium using the sedimentation kernel; ( ii ) in only two cases of agitation ( i.e., 350 and 500 rpm) because for lower values of stirring rates it would be necessary to couple the two mechanisms of turbulent aggregation and aggregation through sedimentation; so far this has not been done. The simulations have given the following main results: (i) The aggregates are fractallike with a fractal dimension Df equal to 2.45 { 0.05 in still medium. In stirred medium, a value of Df Å 2.20 is consistent with the experimental turbidity curves within 10% accuracy. (ii) Fragmentation is not necessary to explain the characteristics of the process. These results are quite consistent with the existence of a cohesion force between primary particles much stronger in n -heptane than in water. We have already mentioned the partial irreversibility of aggregation in n -heptane under ultrasonic action and the presence of aggregates at instant zero. CONCLUSION
Turbidimetry is a good method to follow continuously and for in situ aggregation of alumina in water and n heptane. The turbulent hydrodynamic conditions have an influence on both aggregation and fragmentation of the aggregates. The model proposed here is based on the now-classical theory of turbulent aggregation ( for particles and aggregates smaller than the Kolmogorov micro-
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scale ) . However, it takes into account recent developments concerning the aggregation and breakage of fractallike aggregates. These latter two improvements appear as essential. In particular, a good agreement between measured and predicted turbidity is found for a fractal dimension of 2.30 in water and 2.20 in n -heptane, both under turbulent conditions. These characteristics are in agreement with already-published results and do not seem to depend on the stirring rate.
REFERENCES 1. Kolmogorov, A., Dokl. Akad. Nauk. SSSR 30, 301 (1941). 2. Hinze, J., ‘‘Turbulence.’’ McGraw-Hill, New York, 1975. 3. Landau, L. D., and Lifshitz, F. M., ‘‘Fluid Mechanics.’’ Pergamon Press, Oxford, 1984. 4. Lesieur, M., ‘‘La Turbulence.’’ Presses Universitaires, Grenoble, 1994. 5. Verwey, E. J. W., and Overbeek, J. T. G., ‘‘Theory of the Stability of Lyophobics Colloids.’’ Elsevier, Amsterdam, 1948. 6. Casimir, H. B. G., and Polder, D., Phys. Rev. 73, 360 (1948). 7. Schenkel, J. H., and Kitchener, J. A., Trans. Farad. Soc. 56, 161 (1960). 8. Chapman, D. L., Phil. Mag. 25, 475 (1913). 9. Gouy, G., Ann. Phys. 9, 129 (1917). 10. Smoluchowski, M., Z. Physik. Chem. 92, 129 (1917). 11. Fuchs, N., Z. Phys. 89, 736 (1934). 12. Spielman, L. A., J. Coll. Interface. Sci. 33, 562 (1970). 13. Van de Ven, T. G., and Mason, S. G., Colloid Polym. Sci. 255, 468 (1977). 14. Zeichner, G. R., and Schowalter, W. R., AIChE J. 23, 243 (1977). 15. Higashitani, K., Yamauchi, K., Hosokawa, G., and Matsuno, Y., J. Chem. Eng. Jpn. 15, 299 (1982). 16. Higashitani, K., Yamauchi, K., Hosokawa, G., and Matsuno, Y., J. Chem. Eng. Jpn. 16, 299 (1983). 17. Saffman, P. G., and Turner, J. S., J. Fluid Mech. 1, 16 (1956). 18. Entov, V. M., Kaminskii, V. A., and Lapiga, E. Y., Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 10, 47 (1976). 19. De Boer, G. B. J., Hoedemakers, G. F. M., and Thoenes, D., Chem. Eng. Sci. 67, 301 (1989). 20. Gruy, F., and Saint-Raymond, H., J. Colloid Interface Sci. 185, 281 (1997). 21. Levich, V. G., ‘‘Physicochemical Hydrodynamics.’’ Prentice Hall, New York, 1962. 22. Ayazi Shamlou, P., Stavrinides, S., Titchener-Hooker, N., and Hoare, M., Chem. Eng. Sci. 49, 2647 (1994). 23. Kusters, K. A., Wijers, J. G., and Thoenes D., Chem. Eng. Sci. 52, 107 (1997). 24. Gmachowski, L., J. Colloid Interface Sci. 178, 80 (1995). 25. Torres, F. E., Russel, W. B., and Schowalter, W. R., J. Colloid Interface Sci. 142, 554 (1991). 26. Kusters, K. A., ‘‘The influence of turbulence on aggregation of small particles in agitated vessel.’’ Ph.D. Thesis, Eindhoven University of Technology, The Netherlands (1991). 27. Sonntag, R. C., and Russel, W. B., J. Colloid Interface Sci. 115, 378 (1987). 28. Oles, V., J. Colloid Interface Sci. 154, 351 (1992). 29. Luo, H., and Svendsen, H. F., AIChE J. 42, 1225 (1996). 30. Randolph, A. D., and Larson, M. A., ‘‘Theory of Particulate Processes.’’ Academic Press, New York, 1988.
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31. Abrahamson, J., Chem. Eng. Sci. 30, 1371 (1975). 32. Adachi, Y., Cohen Stuart, M. A., and Fokkink, R., J. Colloid Interface Sci. 165, 310 (1994). 33. Baldi, G., Conti, R., and Alaria, E., Chem. Eng. Sci. 33, 21 ( 1978 ) . 34. Wu, H., and Patterson, G. K., Chem. Eng. Sci. 44, 2207 (1989). 35. Friedlander, S. K., ‘‘Smoke, Dust, Haze,’’ Wiley, New York, 1977. 36. Smitt, D. J., Hounslow, M. J., and Paterson, W. R., Chem. Eng. Sci. 49, 1025 (1994). 37. Van de Hulst, H. C., ‘‘Light Scattering by Small Particles.’’ Wiley, New York, 1957. 38. Kerker, M., ‘‘The Scattering of Light and Other Electromagnetic Radiation.’’ Academic Press, New York, 1969. 39. Elic¸abe, G. E., and Garcia Rubio, L. H., J. Colloid Interface Sci. 129, 192 (1989).
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40. Crawley, G. M. ‘‘Granulome´trie des suspensions de particules fines par mesures turbidime´triques spectrales.’’ Ph.D. Thesis, Ecole des Mines de Saint-Etienne, France (1994). 41. Crawley, G. M., Cournil, M., and Di Benedetto, D., Powder Technol. 91, 197 (1997). 42. Khlebtsov, N. G., Appl. Opt. 35, 4261 (1996). 43. Rannou, P., Cabane, M., Chassefiere, E., Botet, R., McKay, C. P., and Courtin, R., Icarus 118, 355 (1995). 44. Lips, A., and Levine S., J. Colloid Interface Sci. 33, 455 (1970). 45. Caesarano III, J., Aksay, A., and Bleier, A., J. Am. Ceram. Soc. 71, 250 (1988). 46. Saint-Raymond, H., ‘‘Etude de l’agglome´ration par turbidime´trie de poudres d’alumine en milieu liquide.’’ Ph.D. Thesis, Ecole des Mines de Saint-Etienne, France (1995). 47. Spicer, P. T., and Pratsinis, S. E., AIChE J. 42, 1612 (1996). 48. Allain, C., Cloitre, M., and Parisse, F., J. Colloid Interface Sci. 178, 411 (1996).
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CS985436
Turbulent Aggregation of Alumina in Water and n-Heptane Hubert Saint-Raymond, Fre´de´ric Gruy, and Michel Cournil 1 Ecole des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint Etienne Cedex 2, France Received April 15, 1997; accepted January 23, 1998
THEORETICAL BACKGROUND Aggregation of alumina powder in water and n-heptane is followed by turbidimetry in situ. The experiments are carried out under turbulent conditions. They are discussed in the framework of a model which takes into account the hydrodynamic and physicochemical aspects of aggregation. A good agreement between measured and predicted turbidity is found provided the aggregates are considered as noncompact and liable to undergo fragmentation. Recent models of aggregation and fragmentation of fractallike clusters are successfully tested. The aggregate morphology seems to depend on the nature of the liquid medium, however, not on the stirring rate. q 1998 Academic Press Key Words: aggregation; fractal; turbulence; breakage; turbidimetry; light scattering.
Turbulence
In this work, we adopt the classical description of turbulence as presented in (1–4). At any instant and in any place of the turbulent flow, eddies are supposed to appear and to take part in the energy dissipation. Energy is supplied to the flow by an external source, such as mechanical stirring. Energy is transferred from the largest eddies to the smallest eddies in which it is dissipated through viscous interactions. The size of these smallest eddies is the Kolmogorov microscale h; from dimensionless considerations h is expressed as a function of the kinematic viscosity n and the energy dissipation rate per unit mass em in the form
INTRODUCTION
hÅ
Aggregation is one of the most complex steps in powder technology; this complexity is due to the variety of phenomena or aspects involved: hydrodynamics of suspensions, interactions between particles, and morphology of the aggregates. In spite of the existence of several, valuable models applicable to the separate aspects of the phenomenon, so far, very few comprehensive theoretical approaches are able to describe the turbulent aggregation of polydisperse particles in physicochemical and hydrodynamic interaction, particularly if the so-formed aggregates are not compact and can break up. This is the task that we propose to undertake in this paper, using results from a turbidimetric study of aggregation of alumina powders in water and in n-heptane. This paper is organized as follows: after a review of the theoretical background connected with aggregation, we present the experimental data available on alumina; then, the different steps of the model, each corresponding to a theoretical aspect, are discussed and adapted to the present case. Last, theoretical predictions are compared to the experimental results, and a global interpretation is proposed.
1
To whom correspondence should be addressed.
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n3 em
1/4
.
[1]
The velocity gradient gg in each eddy is proportional to ( em / n ) 1 / 2 . The resulting shear stress is ts Å mgg .
[2]
In this expression, m denotes the dynamic viscosity. Interaction Between Solid Particles
Aggregation is the consequence of a collision between particles. The mechanism that brings particles into close proximity results from the hydrodynamics of the suspension; however, the trajectories of the particles just before their encounter, the collision efficiency, and the aggregate cohesion depend on the different interactions between particles. London–Van der Waals Attractive Interactions These forces have their origin in interactions between instantaneous induced dipoles. For two spheres of radii ai and aj separated by a distance H, an attractive interaction potential VA can be calculated (5). In fact, a more correct expression is obtained by taking into account a ‘‘retardation’’ effect (6) due to propagation of electromagnetic waves in a dielec-
238
0021-9797/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.
S D
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tric medium. Schenkel and Kitchener (7) have proposed an empirical correlation to calculate the attractive potential as a function of the geometrical parameters (ai , aj , H) and of A, the Hamaker constant of the particle in its medium. Double-Layer Repulsive Interactions A solid particle located in an electrolyte generally takes a surface charge and is surrounded by a layer of dissolved ions. Several models exist to describe this double layer (5, 8, 9). The zeta potential is the only characteristic parameter of this double layer which is experimentally available; it is currently assimilated to the surface potential c0 in the case of dilute liquid media. When two particles get into contact, the interactions of their respective double layers result in a repulsive potential VR . In this paper, only experiments for which the double-layer interactions were zero or negligible are presented. For this reason, we do not give further details on this subject. Hydrodynamic Interactions The problem is to describe the conditions of collision of two interacting particles brought into close proximity by a viscous fluid. Following the pioneering works of Smoluchowski ( 10 ), several authors ( 11, 12) expressed the Brownian aggregation flow Jij ( R) of particles of radius aj which collide with the sphere of radius R surrounding a reference particle of radius ai : Jij Å 04pR 2G(R) 1
F
(Di / Dj )
ÌNj Nj ai / aj ÌVT / ÌR 6pm ai aj ÌR
G
.
[3]
G(R), the corrected mobility coefficient, is a function of R, ai , and aj ; Di is the Brownian diffusion coefficient of a sphere of radius ai Di Å
k BT , 6pmai
[4]
where kB is the Boltzmann constant, T is the temperature, Nj is the number of particles of radius aj per unit volume, and VT is the total interaction potential. The effect of these interactions is characterized by the stability factor Wij , defined as Wij Å
J 0ij (R) , Jij ( R)
Wij which takes into account the attractive and repulsive interactions and also the effect of the viscosity via G(R). The models derived from Eq. [3] assume that aggregation takes place through Brownian motion. Brownian aggregation is important for submicrometer particles; however, when their size becomes larger, the particles are submitted to the action of the flow and other aggregation mechanisms take place. More recent studies (13–16) take into account the different types of interactions in a shear flow. Their authors introduce the capture efficiency coefficient aij between two particles of radius ai and aj ; aij is the ratio of the calculated particle collision flow to the flow value predicted by Smoluchowski ( aij Å W 01 ij ). The model of Van de Ven and Mason (13) includes attractive, repulsive, and hydrodynamic interactions in the case of equally sized particles, whereas the model of Higashitani et al. (15, 16) can be applied to any pair of spherical particles, in the case of attractive and hydrodynamic interactions only. The case of aggregation in turbulent flows has been little investigated; two tracks have been explored so far. For particles smaller than the Kolmogorov microscale, aggregation is assumed to take place in the smallest eddies under local shear flow conditions. This makes possible the use of the previous models (15–19). Most recently, two of us (20) have proposed the use of the Spielman equation (Eq. [3]) by replacing the Brownian diffusion coefficient Di with the Levich (21) turbulent diffusion coefficient Dturb . We proposed an analytical expression for aij in the most general case of interactions and radii values in a turbulent flow. Collision Efficiency for Noncompact Aggregates
The morphology of the aggregates depends on both the physicochemical and the hydrodynamic conditions of their formation as well as on their intrinsic mechanical properties. However, the aggregation dynamics also depend on the morphology of the colliding particles. Recent experiments have shown that aggregates have a fractal structure (22, 23). An aggregate containing i primary particles of radius a1 is characterized by the fractal dimension Df , the outer radius ai , and the hydrodynamic radius aH . Because the structure of the aggregates is nonuniform, their volume density f(r) depends on the distance r from the center of mass of the aggregate, where the average volume density is denoted fU . These different characteristics are linked by the relations (24, 23)
[5]
ai Å a1
where J 0ij (R) denotes the Jij ( R) value in the absence of interactions. Spielman (12) has proposed a determination of
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f(r) Å
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SD SD i S
1/D f
S r Df 3 a1
[6] D f 03
[7]
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SAINT-RAYMOND, GRUY, AND COURNIL
SD
fU Å S
ai a1
D f 03
,
[8]
where S is a structure factor. The authors differ in the way they express the ratio between the hydrodynamic radius and the outer radius; this ratio is assumed to be constant by (22, 25), whereas, according to (23), it depends on the aggregate volume density. However, for fractal dimensions close to 2, the two models are in agreement. From computer simulations, Gmachowski (24) has found the following relation between S and Df : S É 0.42 Df 0 0.22.
[9]
We present here the procedure used by Kusters (23, 26) (shell-core approach) to calculate the collision efficiency aij between two porous aggregates (i ¢ j ú 1). (i) Determination of the permeability of the bigger particle (26): 30 ki Å
9 fU 2
1/3
/
S
9 fU 2
9fU 3 / 2fU
05 / 3
D
5/3
0 3fU
2
2a 21 ,
[10]
CU
where CV is an average shielding coefficient. (ii) Determination of the Debye shielding ratio of this particle: jÅ
ai . k 1/2
[11]
(iii) Determination of the collision efficiency (Kusters model): a iK, j as a function of j; from the results given in (26), the following relations can be derived after curvefitting for aj ú 0.1: ai
a iK, j ( j ) Å 1 0 tanh 0.1 j 1.35 ,
[12a]
aj õ 0.1: ai
a iK, j ( j ) Å 1 0 tanh 0.18 j 0.4 .
[12b]
(iv) Determination of the collision efficiency a iH, j (Higashitani model) for compact aggregates (15, 16). (v) Comparison between a iK, j and a iH, j and choice of the ai , j value. The shell-core model takes into account the porous fractal structure of the aggregates but ignores the interactions other
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than hydrodynamic, whereas, within the Higashitani approach, hydrodynamic and attractive interactions are considered between compact aggregates. Somewhat to take advantage of the most interesting aspects of each model, it is proposed in (23) that ai , j be chosen in the following way: if a iK, j õ a iH, j , then ai , j Å a iH, j , else ai , j Å a iK, j . Break-Up of Aggregates
In the aggregation processes, a maximum aggregate size is almost always observed (19, 26, 28). This can be due to several reasons: breakage, collision efficiency becoming zero beyond a critical size, or settling which could remove the largest aggregates from the system. The second cause is unlikely in the case of noncompact aggregates, and the third one requires particle sizes much larger than the maximum size reached experimentally. Breakage is certainly the main cause of this limit in size. The occurrence of breakage depends on the balance between the desaggregation effects due to the action of the fluid and the overall cohesion of the aggregate due to the interactions between primary particles. The hydrodynamic effects are of different nature depending on whether the aggregate is larger or smaller than the Kolmogorov microscale. Only the latter case is compatible with the experimental conditions of this study. It corresponds to a shear stress originating from the local velocity gradient and acting on the aggregate. The authors do not agree on the way to express the competition between the desaggregation and the cohesion effects. According to the models, the relevant parameter is either the ratio Ec /Et or the ratio s / ts ; Ec and Et are, respectively, the aggregate cohesion energy and the turbulent energy acting on this aggregate; s is the mean mechanical strength of the aggregate; and ts is the mean shear stress (22, 26, 27, 29). The breakage rate is proportional respectively to gg e 0E c /Et and to gg e 0s/ts . Another discussion is relative to the size of the fragments produced by breakage. Two cases are currently envisaged: • Erosion of single or small groups of particles from the aggregate surface (22). • Production of fragments of comparable sizes (26, 27).
In all cases, the breakage rate depends on the hydrodynamic conditions of the flow, via em and m, and on the characteristics of the aggregates—outer radius, fractal dimension, primary particle radius, and cohesion force between two primary particles. This force F is expressed in the form FÅ
Aa1 . 12H 20
[13]
H0 is the separation between two primary particles in the aggregates.
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Aggregation Kernels
The variation of the volume density Ni of aggregates Ai versus time t is given by the population balance equation in its discrete form (30) for i ú 1:
dNi 1 Å dt 2
∑ Kj,i0j Nj Ni0 j 0 ∑ Ki ,k Ni Nk j Å1,i01
0
i Å 1:
1 2
kÅ1,`
∑ K if, j Ni / ∑ K j,if Nj j Å1,i01
[14]
j úi
dN1 f Å 0 ∑ K1,k N1 Nk / ∑ K j,1 Nj , dt kÅ1,` j ú1
[15]
where Kij is the aggregation kernel [i-mer / j-mer r (i / j)-mer] and K if, j the breakage kernel [i-mer r j-mer / (i 0 j)-mer]. According to the type of flow and the particle size, Brownian kernels or turbulent kernels can be considered. The relevant parameters are b1 , the ratio of the particle size to the Kolmogorov microscale, and b2 , the ratio of the particle size to the Levich critical diameter dc (21). For b1 ú 1, models of highly turbulent suspensions should be applied (31). For b1 õ 1 and b2 ú 1, the turbulent models (which in fact assume a local shear flow in the Kolmogorov’s eddies) can be used. For b1 õ 1 and b2 õ 1, the Brownian contribution is no more negligible. In this case, Adachi et al. (32) propose that Kij be expressed as the sum of two contributions Kij Å (Kij )Br / (Kij )turb
[16]
in which (Kij )Br is the Brownian kernel given by Smoluchowski and possibly corrected by the stability factor Wij (equal to 2 in the experimental work of Adachi) and the turbulent kernel (Kij )turb is derived from one of the turbulent models presented earlier. For instance, in the case of particles smaller than the Kolmogorov microscale, the following expression is currently used: (Kij )turb Å
4 3
S D SD 3p 10
1/2
em n
1/2
d (ai / aj ) 3 aij . [17]
This relation comes from the Saffmann-Turner approach (17); em is the mean energy dissipation rate; d is a correction coefficient which is introduced to take into account different deviations from this ideal model. Many expressions are found in the literature for the mean value of em . For instance, Baldi et al. (33) give Np v 3 D 5s , V
[18]
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eU m Å
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where Np is the power number, Ds is the stirrer diameter, v is the rotation rate of the stirrer, and V is the volume of the suspension. This type of expression should be used with precaution because, except for highly turbulent media, em is not uniform in a stirred vessel. In particular, Wu and Patterson (34) show that, in a baffled stirred tank with a sixblade turbine, the dissipation in the impeller region is about 15 times as much as that in other parts of the tank; the fluid in this region represents only 9% of the total. The consequence of the turbulence heterogeneity can be taken into account via d [ for instance, d Å 0.7 in (26)]. Particles in an unstirred medium can form aggregates while settling (35, 36). Different aggregation kernels have been proposed according to the particle size concerned. For small sizes (ai õ 50 mm), the terminal velocity of a particle is given by the Stokes law and is proportional to a 2i ; hence, the expression of the aggregation kernel for gravitational settling is Kij Å
2p ( rs 0 rL )g (ai / aj ) 2Éa 2i 0 a 2j É . m 9
[19]
In this relation, rs and rL are the respective densities of the solid and liquid phases; g is the gravity acceleration. Turbidimetry
General Scope In this work, aggregation of alumina powders has been studied using an in situ turbidimetric sensor. This device allowed us to record continuously a signal directly dependent on the particle size distribution (PSD) in the reactor and so we avoid two drawbacks: • Sampling from the suspension, which could damage the aggregates and definitely introduces systematic counting errors, in particular concerning the finest particles which are difficult to collect. • A bypass through the measurement cell of a commercial particle sizer, with again the risk of damaging the aggregates by pumping and definitely by uncontrolled hydrodynamical disturbances.
The turbidity of a polydisperse suspension of spheres of diameter d is linked to the PSD by the Mie theory (37, 38), t( l ) Å
*
`
Csca f ( d)dd,
[20]
0
where t( l ) denotes the turbidity at wavelength l; f ( d), the population density; and Csca , the Mie scattering cross-section. The inverse problem (i.e., the determination of the PSD from the measured turbidity spectrum) is acknowledged to
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FIG. 1. Schematic diagram of the experimental setup.
be an ill-conditioned mathematical problem which can be solved by the use of constrained least-squares methods (39). In recent works, we defined the interests and limits of such a method in particle sizing (40, 41). For instance, a low volume fraction f of solid is required, generally f õ 10 04 for micrometer particles. Due to its ill conditioning the inverse problem can be solved in good conditions only if the main part of the sample belongs to the diameter range 0.1– 10 mm. Nevertheless, the direct calculation of turbidity spectra, as expressed by relation [20], is valid for all particle diameters.
Most recently, a procedure of determination of light scattering and extinction by an ensemble of fractal clusters was proposed by Khlebtsov (42) and validated for a system consisting of aggregating polystyrene latex particles. However, due to the size of the primary particles (90 nm) , the approximation of Rayleigh-Gans —here approximately equivalent to a1 ! l — was used by Khlebtsov in its work and the aggregate scattering cross-section Csca,R0G was calculated in this theoretical framework. For larger primary particles, not too far from the Rayleigh-Gans domain, it has been suggested (43) to calculate Csca , the actual value of the scattering cross-sections by correcting Csca,R0G in the following way: C 1sca C 1sca,R0G
.
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Experimental Setup, Experimental Procedure and Products
(i) Measurement set of turbidity. The light source is a 75-W high-pressure Xe–Hg arc lamp which emits a polychromatic light in the UV–visible range (190–800 nm). Light is passed to the sensor window through optical fibers
[21]
In this relation, C 1sca and C 1sca,R0G are both the scattering cross-sections of a primary particle; however, the former has
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The experimental set-up used in this study can be divided into three parts (Fig. 1).
Turbidity of Aggregates
Csca Å Csca,R0G
been calculated using the rigorous Mie theory, whereas the latter is obtained from the Rayleigh-Gans approximation. The Khlebtsov procedure has been conceived for aggregates of many particles. For doublets, for instance, the notion of fractal dimension has no meaning, of course; however, the optical basis of the Csca calculation can be kept (44).
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FIG. 2. Top view of the stirred vessel (fitted with a turbidimetric sensor ) .
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FIG. 3. Turbidity variation throughout a typical experiment.
600 mm in diameter. The window length is the optical path which ranges from 0.1 to 17 mm. Each of the sensor window ends is fitted with a convergent lens which ensures the beam collimation in the measurement zone. After the light beam has crossed the sensor window, scattering by fine particles results in the attenuation of its intensity. The transmitted light is again passed via optical fibers to a spectrophotometer which is equipped with a monochromator (holographic diffraction grating) and a light detector consisting of a 35 photodiode array. This setup can deliver continuously the turbidity spectrum of the suspension. (ii) Reactor. The reactor (Fig. 2) is a glass cylindrical vessel with diameter and height both equal to 10 cm. It is double-jacketed to ensure the temperature control. The turbidity sensor is located at 3 cm above the bottom. Stirring is operated by a four-blade 457 impeller that is 4.6 cm in diameter and located at the same height as the sensor. To ensure a good dispersion of the solid phase at the beginning of each experiment, a ultrasonic probe can be used. (iii) Data acquisition and processing set. Each available turbidity spectrum is in fact the average of 20 spectra recorded in the total time of 3 s. This allowed us to follow the evolution kinetics of relatively rapid processes over a period of several hours.
FIG. 4. Initial particle size distribution (PSD) of alumina in water.
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243
FIG. 5. Zeta potential of alumina in water at different pH values.
Each experiment consists of five different steps which are represented in Fig. 3: ( i ) Filling the reactor with 500 cm3 of the desired liquid medium; starting thermal control and stirring ( zone I in Fig. 3 ) . (ii) Measuring of the blank turbidity signal (i.e., in absence of solid). (iii) Introducting the solid sample (10–100 mg); natural dispersion in the liquid medium during a 5-min period (zone II in Fig. 3). (iv) Starting the ultrasonic action for a 5 min time (zone III in Fig. 3). (v) Ending the ultrasonics action; this instant is taken as zero time (t 0 in the figures) for the free evolution of the system (zone IV in Fig. 3). In this work, aggregation has been studied on 50-mg samples of alumina powders from Baikowski Company. This alumina
FIG. 6. Turbidity variation with time during aggregation for different wavelengths (50 mg alumina, pH Å 9.5, stirring rate Å 350 rpm).
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FIG. 7. Turbidity spectra at different times during aggregation (50 mg alumina, pH Å 9.5, stirring rate Å 350 rpm).
consists at least of 95% a-alumina with a BET specific surface area of 5.7 m2rg 01 . Figure 4 shows the PSD by number obtained from measurements in a Coulter LS 130 laser particle sizer. Immediately before the PSD determination, the sample was dispersed ultrasonically. A large number of particles have diameters close to 0.2 mm, whereas fewer aggregates with diameters ranging from 1 to 6 mm are observed. This is confirmed by microphotographs. Figure 5 shows the zeta potential values of alumina in water measured on a Pen Kenn Laser Zee Meter at different pH values. High absolute values of zeta potential are currently associated with strong repulsive interactions and consequently to weak aggregation, whereas for low values— say smaller than 20 mV, corresponding to the pH zone 8– 9.5—aggregation is assumed to be easier. Most of the experiments reported here have been performed for pH values belonging to this interval. The zero value of zeta potential
FIG. 8. Reversibility of aggregation (US Å ultrasonics action).
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FIG. 9. Influence of pH on turbidity variation during alumina aggregation in water (350 rpm; l Å 501 nm).
is reached for pH 9. This result is in agreement with the available experimental data (45). Results
The available results from each experiment can be put in two forms: t(t) for different wavelengths (Fig. 6) or t( l ) at different times (Fig. 7). Because little will be drawn from the turbidity spectra, we will comment mainly on t (t) plots obtained at a given wavelength value. Experiments in Water Reversibility of the aggregation process. Figure 8 shows the effect of ultrasonics on aggregation of alumina at pH
FIG. 10. Influence of stirring rate on turbidity variation during alumina aggregation in water (pH Å 9.5; l Å 501 nm).
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sufficient to breakup the aggregates and to recover the initial state. DISCUSSION
Qualitative Discussion
FIG. 11. Influence of stirring rate on turbidity variation during alumina aggregation in n-heptane ( l Å 501 nm).
9.5 and a stirring rate of 350 rpm. After the final t value has been reached, ultrasonics are applied once again. After their action period, the same values of initial turbidity and the same time evolution as previously are observed. These experiments tend to prove that the aggregation of alumina in water is widely reversible: • The aggregates can be destroyed in their elementary components under the action of a sufficient mechanical stress. • The identity of the primary particles is not affected by aggregation, since they can apparently be recovered with the same properties.
The continuous decrease in turbidity as aggregation proceeds can be essentially explained by the decrease in particle number (Eqs. [14], [15], and [20]). The effect of the stirring rate is certainly to increase the frequency of collisions between particles and/or aggregates. The fact that t(t) does not tend asymptotically to zero could be explained by the existence of a maximum aggregate size. This important point will be examined further. In acid (pH õ 6.7) or basic media (pH ú 10), aggregation is very weak. In these two pH zones, the absolute value of zeta potential (Fig. 5) is greater than 30 mV; this corresponds to a strong repulsive interaction. In contrast, in the pH zone, 8.5–9.5, the very low values of zeta potential allow aggregation to occur. The maximum aggregation rate is observed for pH 9.5, that is, for a nonzero zeta potential value, and namely equal to 020 mV. This is hardly understandable in the framework of the classical theories which consider that a nonzero potential results in repulsive forces between particles and reduces aggregation. Because the measurements of zeta potential have performed under experimental conditions very similar to the present ones, the only explanation for our observation is that, in our case, the zeta potential cannot be assimilated to the surface potential, contrary to
Influence of pH. The pH of the aqueous solution has been varied by addition of hydrogen chloride or sodium hydroxide solutions. Figure 9 shows the different t(t) plots obtained at 350 rpm and at 501 nm at different pH values. In acid media, turbidity varies little. In the pH domain 8.5– 9.5, a sharp turbidity drop occurs at early times. For pH values larger than 10, the turbidity varies very slowly. In what follows, we will focus our interest on the experiments performed at the pH value of 9.5, because it corresponds to the most rapid variation in the turbidity. Influence of the stirring rate. The influence of the stirring rate is presented for a pH value of 9.5, a mass of 50 mg, and a wavelength of 501 nm (Fig. 10). Results for other pH values are presented in (46). Experiments in n-Heptane A turbidity decrease with time is observed too for alumina suspensions in n-heptane. The effect of the stirring rate is shown in Fig. 11. In contrast to the behavior observed in water, aggregation in n -heptane is not an reversible phenomenon; in particular, the action of the ultrasonics is not
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FIG. 12. Normalized scattering cross-section of fractal aggregates (fractal dimension Df ; number of primary particles in an aggregate n).
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the usual assumptions. Thus we will consider that the surface potential is zero in n-heptane as well as in water at pH 9.5. Because of this unexpected difficulty and also because no model is available to represent aggregation between porous grains in the presence of repulsive electrostatic interactions, we do not deal in this paper with the interpretation of the influence of pH on aggregation.
Models
• The advantage of a not too small wavelength value is to keep the system not too far off the validity range of the models of light scattering by fractal objects (Rayleigh-Gans domain for the primary particles).
In both respects the choice of the value of 501 nm appears a good compromise. Turbidity spectra at zero time. The PSD of alumina in water as obtained from the Coulter LS130 sizer measurements is bimodal and characterized by two peaks of respective mean diameter and geometrical standard deviation:
Turbidity Turbidity calculations. The solid samples studied here contain individual particles of diameter d1 Å 0.2 mm and larger aggregates, which probably consist of these primary particles. The wavelength range investigated is 300–800 nm. Thus, all the particles and aggregates are outside the Rayleigh-Gans domain (d ! l ); however, they are not very far from its upper validity limit. Consequently, the different scattering cross sections have been calculated according to the models presented in the introduction of this paper: • Mie rigorous theory for the primary particles; • approach of Lips and Levine (44) for the doublets; • corrected Khlebtsov procedure for larger aggregates.
Figure 12 shows the results of the calculations for different values of the fractal dimension and the number of particles in the aggregates. To emphasize the effect of the aggregate fractal structure, the scattering cross section (Csca, f ) has been made dimensionless through normalization by the corresponding scattering cross section of a compact object (Csca,c ) with the same number of primary particles. Using this representation, it clearly appears that a ramified object can scatter much more light than a compact object, and this occurs the more ramified and large it is. Once calculations for these scattering cross-sections have been done, the turbidity of a given population of particles and aggregates can be determined from Eq. [20]. Exploitation of the turbidity data. In the present work, the inversion procedure did not successfully calculate the PSD from the measured turbidity spectrum probably because of the broad PSD observed as early as the beginning of aggregation. However, relation [20] has been widely exploited to calculate the turbidity spectra corresponding to PSD derived from the different aggregation models. Thus validity of the models has been judged from the comparison between predicted and measured turbidity values. These comparisons have been often performed at the value of 501 nm in wavelength for two reasons: • It is one of the wavelength values for which the turbidity drop was always significant (in contrast to other values such as 800 nm).
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dU 1 Å 0.19 mm; s1 Å 0.42 (primary particles) dU 2 Å 1.3 mm; s2 Å 0.25 (aggregates). The ratio of the number of aggregates to the number of primary particles is: N2 /N1 Å 5.6 1 10 04 . Two problems arise about these data; the problems concern the representativeness of the data to describe the initial state of the aggregating powder: j
The PSD characteristics are obtained from a commercial software which probably ignores the refinements of fractal structures. j The measurements have not been performed in line, although the solid samples have been dispersed ultrasonically. The former reservation can be easily objected to because, due to the low number of aggregates in the sample, the calculated turbidity spectrum depends only slightly on the fractal dimension. Concerning the second point, we have found that the calculated turbidity spectrum is in agreement with the experimental data within 10% accuracy. Thus we can conclude in favor of the representativeness of the sizer measurements. Because no experimental PSD of alumina in n-heptane was available, we estimated it through curve-fitting of the experimental turbidity spectrum to obtain the bimodal distribution: dU 1 Å 0.19 mm; s1 Å 0.32 (primary particles) dU 2 Å 1.3 mm; s2 Å 0.25 (aggregates) and
N2 Å 1.13 1 10 03 . N1
The corresponding calculated turbidity spectrum is in agreement with the measured spectrum within 5% accuracy. The larger proportion of aggregates is likely due to stronger cohesion forces and consistent with the not completely reversible aggregation observed in n-heptane. Aggregation Model: Main Assumptions Models of aggregates. The aggregate fractallike structure is described using the relations [6] – [9]; it is character-
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ized by the parameters d1 (d1 Å 0.2 mm) and Df . The fractal dimension has been assumed constant throughout an aggregate, whatever its size, and over a whole experiment. However, it has been considered as a priori depending on the experimental conditions, in particular, stirring rate and nature of the dispersing medium. The problem of the determination of H0 will be discussed later. Aggregation kernels. The turbulent kernels have been calculated from relation [17 ] using the method proposed by Kusters and recalled earlier. The nonuniformity of the energy dissipation rate in the reactor has been taken into account using the same value of the correction coefficient as we did. This appeared to us to be a reasonable assumption, considering the analogies between the two reactors. In these calculations, the power number Np has been taken equal to 1.2, a value commonly assumed for a four-blade 457 impeller ( 33 ) . The global aggregation kernel (i.e., including the turbulent and Brownian contributions) has been determined following the Adachi relation (Eq. [16]). As we said before, no repulsive interaction has been considered. In our experimental system, the Hamaker constant is equal to 2.67 1 10 020 J in water and 2.38 1 10 020 J in n-heptane; from this value, the Brownian stability factor Wij has been found to be close to 2 in most cases. The kernel of aggregation in a still medium has been calculated using relation [19]. Fragmentation kernel. In this paper, we have successively tested the assumptions of fragmentation of Ayazi Shamlou et al. ( 22 ) ( surface erosion ) and of Kusters ( 26 ) ( breakage in fragments of comparable sizes ) . In each case the corresponding fragmentation kernel K if, j has been calculated. The erosion model has been used without modification; its parameters are H0 and Df . Concerning the second model, we have retained its main assumption, that is, the possibility of breakage in fragments. However, we have envisaged several fracture ratios: i r 2 ( i / 2 ) , i r ( 3 i / 4 / i / 4 ) , . . . and adopted the way of calculation of Ayazi Shamlou et al. ( 22 ) which seemed us to be more satisfactory from a physical point of view. In the same way as in the calculations of the aggregation kernels, we took into account the nonuniformity of the turbulence; consequently, the fragmentation kernel K if, j has been obtained by averaging : gg e 0 s / ts . In the case of fragmentation too, H0 and Df , as well as the fracture ratio, must be estimated.
gated, that is, 0.2–20 mm, may require excessive computation time. To avoid this difficulty, we have adapted a discretization method proposed by Spicer and Pratsinis (47) and known as particularly efficient and rapid in the coagulationfragmentation problems. The size domain is divided in adjacent sections numbered by index i. To the ith section is associated a volume Vi that is the average volume of the aggregates contained in this section, Vi Å
bi / bi01 , 2
where bi is the upper boundary volume of section i; the volume of section i is obtained from the volume of the previous section i 0 1 by the relation Vi Å 2Vi01 (V1 corresponding to the volume of a primary particle). In this study, the discretization in diameter has been achieved using 20 sections. The so-obtained ordinary differential equations system has been solved using the explicit Euler method (which revealed as efficient as the RungeKutta algorithm, currently used in this sort of problems). For each calculated PSD, the corresponding turbidity can be determined using Eq. [20]. The validation of the different models and the estimation of the unknown parameters were carried out from the comparison between experimental and simulated t(t) at a given wavelength, here 501 nm. More
Numerical Methods The evolution of the particle size distribution is simulated by solving the population balance equations [14, 15]. The necessary discretization over the wide size range investi-
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FIG. 13. Calculated turbidity variation with time as a function of the fractal dimension Df of the aggregates (exp.: experimental plot at 300 rpm and l Å 501 nm; aggregation in water).
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precisely, each of these curves has been first made dimensionless by division through the initial turbidity value t(0). Simulation Results for Aggregation in Water
Determination and Discussion of the Relevant Aspects of the Aggregation Process Interest of the fractallike aggregate models. In order to judge the opportuneness of using fractal aggregation models, we present, in Fig. 13, t(t) plots calculated for different values of the fractal dimension Df . No fragmentation has been considered in these simulations. Only the results obtained for the alumina-water system at the stirring rate 300 rpm (Fig. 13) are discussed here. In the same figure, the experimental plot has been reported too. It is quite clear that aggregation proceeds faster when the fractal dimension decreases and that the process experimentally observed is much faster than the prediction derived from a model of nearly compact aggregates (Df Å 2.95). From comparison between the initial parts of the predicted and observed turbidity curves, the fractal dimension Df has been estimated to the value of 2.30. A most interesting feature in these turbidity plots is their nonzero asymptotic value. This apparent stop in the system evolution is currently attributed to the effect of breakage; however, in these simulations, breakage has still not been considered. Although aggregation proceeds, the turbidity signal no longer varies. The origin of this behavior is in the properties of light scattering of the large fractal aggregates. Nevertheless, this effect is not sufficient to explain the height of the final turbidity level; thus, breakage must be considered too. In the absence of stirring, calculations show a very slow change in turbidity, in agreement with the experimental observations. Influence of the break-up phenomenon. We have first examined the influence of break-up via erosion for a stirring rate of 300 rpm, assuming a fractal dimension of 2.30. The best agreement between calculated and experimental curves was found for H0 Å 8 nm. Then, the breakage of the aggregates in fragments of several size ratios has been envisaged. In all cases, the influence of the fracture ratio reveals as weak. In contrast, the fragmentation phenomenon is strongly dependent on H0 , at least, beyond a minimum threshold (to H0 Å 0 nm and H0 Å 0.5 nm, nearly the same plots). This effect is shown in Fig. 14, which is relative to Df Å 2.30. The best agreement with the experimental curve is found for H0 Å 1.5 nm. In several studies of fragmentation, H0 is often considered as close to 1 or 2 nm (22). Consequently, from the comparison between the two fragmentation models, it appears that the value of H0 predicted by the latter (breakage in two fragments of comparable sizes) is more realistic.
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FIG. 14. Calculated turbidity variation with time as a function of the separation H0 between primary particles in the aggregates; fractal dimension Df Å 2.30 (exp.: experimental plot at 300 rpm and l Å 501 nm; aggregation in water).
Identification of the Model Parameters Starting from the previous considerations and estimations, we have determined the values of Df and H0 for which the best agreement is found between the calculated and experimental t(t ) plots. It appears clearly that both the fractal dimension Df of the aggregates and the separation H0 are nearly constant whatever the stirring rate. The best agreement is found for the already determined values H0 Å 1.5 nm and Df Å 2.30. The most recent turbulent aggregation models (15, 16, 32) are globally in agreement with the experimental results provided they take into account two modifications: ( i ) Assumption of fractallike aggregates ( Df É 2.3 ) . This confirms the results of Kusters et al. ( 23 ) . The fractal dimension in a stirred vessel is equal to 2.30 { 0.05, independent of the stirring rate. According to ( 28, 23 ) , this relatively high value of the fractal dimension is indicative of a restructuring of the aggregates due to low interaction forces between primary particles. This restructuring has been also observed by Allain et al. ( 48 ) for aggregation during settling. ( ii ) Aggregates breakage in two fragments of comparable sizes. Simulation Results for Aggregation in n-Heptane
Alumina aggregation in n-heptane differs from aggregation in water on several points:
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• The process seems to be more rapid in n-heptane and particularly at low stirring rate. • Aggregation occurs at a significant rate even in still medium. This points to a mechanism of aggregation through sedimentation. Compared to water, n-heptane is both less dense and viscous, making sedimentation easier. • At low or moderate stirring rate, the turbidity plots present some similarities of shape with the plot obtained for still medium. This could indicate that sedimentation plays an important part in agitated systems too. • For higher stirring rates, this similarity is less visible. This means that the turbulent aggregation mechanisms have now a major influence on the process. • The final plateau of the turbidity plots is lower than previously found. This could indicate a lesser importance of fragmentation, but is quite consistent with the observation of incomplete reversibility of aggregation in n heptane.
As previously stated, simulations have been performed to validate the different models and determine the characteristics of the aggregates. Turbidity plots have been calculated: ( i ) in the case of the still medium using the sedimentation kernel; ( ii ) in only two cases of agitation ( i.e., 350 and 500 rpm) because for lower values of stirring rates it would be necessary to couple the two mechanisms of turbulent aggregation and aggregation through sedimentation; so far this has not been done. The simulations have given the following main results: (i) The aggregates are fractallike with a fractal dimension Df equal to 2.45 { 0.05 in still medium. In stirred medium, a value of Df Å 2.20 is consistent with the experimental turbidity curves within 10% accuracy. (ii) Fragmentation is not necessary to explain the characteristics of the process. These results are quite consistent with the existence of a cohesion force between primary particles much stronger in n -heptane than in water. We have already mentioned the partial irreversibility of aggregation in n -heptane under ultrasonic action and the presence of aggregates at instant zero. CONCLUSION
Turbidimetry is a good method to follow continuously and for in situ aggregation of alumina in water and n heptane. The turbulent hydrodynamic conditions have an influence on both aggregation and fragmentation of the aggregates. The model proposed here is based on the now-classical theory of turbulent aggregation ( for particles and aggregates smaller than the Kolmogorov micro-
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scale ) . However, it takes into account recent developments concerning the aggregation and breakage of fractallike aggregates. These latter two improvements appear as essential. In particular, a good agreement between measured and predicted turbidity is found for a fractal dimension of 2.30 in water and 2.20 in n -heptane, both under turbulent conditions. These characteristics are in agreement with already-published results and do not seem to depend on the stirring rate.
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31. Abrahamson, J., Chem. Eng. Sci. 30, 1371 (1975). 32. Adachi, Y., Cohen Stuart, M. A., and Fokkink, R., J. Colloid Interface Sci. 165, 310 (1994). 33. Baldi, G., Conti, R., and Alaria, E., Chem. Eng. Sci. 33, 21 ( 1978 ) . 34. Wu, H., and Patterson, G. K., Chem. Eng. Sci. 44, 2207 (1989). 35. Friedlander, S. K., ‘‘Smoke, Dust, Haze,’’ Wiley, New York, 1977. 36. Smitt, D. J., Hounslow, M. J., and Paterson, W. R., Chem. Eng. Sci. 49, 1025 (1994). 37. Van de Hulst, H. C., ‘‘Light Scattering by Small Particles.’’ Wiley, New York, 1957. 38. Kerker, M., ‘‘The Scattering of Light and Other Electromagnetic Radiation.’’ Academic Press, New York, 1969. 39. Elic¸abe, G. E., and Garcia Rubio, L. H., J. Colloid Interface Sci. 129, 192 (1989).
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